Q. 54.2( 484 Votes )

# In Fig. 6.17, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ ROS =(∠ QOS –∠ POS).

Given:  OR is perpendicular to line PQ

To prove:  ROS = (QOS - POS)

Proof:

Now, according to the question,

POR = ROQ = 90°   ( ∵ OR is perpendicular to line PQ)

QOS = ROQ + ROS = 90° + ROS   ............eq(i)

We can write,

POS = POR - ROS = 90° - ROS    ...............eq(ii)

Subtracting (ii) from (i), we get

QOS - POS = 90o + ROS – (90° - ROS)

QOS - POS = 90o + ROS – 90° + ROS

QOS - POS = 2ROS

Hence, proved

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